I can't resist giving an example, from my own book, on what I think "Critical Thinking" in education could look like.

In essence it is not just accepting the imparted knowledge, but examining it from every possible angle, making sure that we know what it is we are talking about.

Sample from my book:

"I always had a problem with Newton's Second Law: F = m * a

We have an equation with three values in it. I know one of these three: I know what acceleration is and how to measure it.

I have an instinctive ‘feel’ for the concepts of mass and force, but I have no idea how to measure them and without measuring instructions, no concept in Physics has any practical value (an essential part of most experiments in Physics is the measurement of the values that play a role in the observed process).

How do we measure this inertial mass: m and force: F?

Any time I asked my teachers what mass was, I was told that it is the measure of an object’s inertia (force divided by acceleration). And when I asked what force was, I was again told, predictably, that it was mass times acceleration.

Finally, years later, I read a few books that satisfied my curiosity. One of the best: Richard Feynman’s “Lectures on Physics” recognizes the validity of my question:

“Let us ask, "What is the meaning of the physical laws of Newton, which we write as F = ma? "What is the meaning of force, mass, and acceleration?" Well, we can intuitively sense the meaning of mass, and we can define acceleration if we know the meaning of position and time. We shall not discuss those meanings, but shall concentrate on the new concept of force. The answer is equally simple: "If a body is accelerating, then there is a force on it." That is what Newton's laws say, so the most precise and beautiful definition of force imaginable might simply be to say that force is the mass of an object times the acceleration.”

Then Feynman states the same problem I was struggling with at University:

“If we have discovered a fundamental law, which asserts that the force is equal to the mass times the acceleration, and then define the force to be the mass times acceleration, we have found out nothing.…Now such things certainly cannot be the content of physics, because they are definitions going in a circle….. One might sit in an armchair whole day long and define words at will, but to find out what happens when two balls push against each other or when a weight is hung on a spring, is another matter al¬together, because the way the bodies behave is something completely outside any choice of definitions.”

Finally, Feynman tells us the solution to the dilemma:

“The real content of Newton’s laws is this: that the force is supposed to have some independent properties in addition to the law F=ma; but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore the physical law F=ma is an incomplete law.”

As far as “what use is Newton’s Law” is concerned, Feynman states that:

“In order to use Newton’s laws, we have to have some formula for the force; these laws say pay attention to the forces. If an object is accelerating, some agency is at work; find it. Our program for the future of dynamics must be to find the laws for the force. Newton himself went on to give some examples. In the case of gravity he gave a specific formula for the force.”

When I read all this, my philosophical problems were solved: Now I knew what Newton meant and how to use his law to solve practical problems. Once you substitute the formula for the force (be it gravity, tension in springs, etc) then I could easily determine the motion of any object subjected to the force if I knew the mass of the object. "

(07-02-2013 07:31 PM)Zat Wrote: I can't resist giving an example, from my own book, on what I think "Critical Thinking" in education could look like.

In essence it is not just accepting the imparted knowledge, but examining it from every possible angle, making sure that we know what it is we are talking about.

Sample from my book:

"I always had a problem with Newton's Second Law: F = m * a

We have an equation with three values in it. I know one of these three: I know what acceleration is and how to measure it.

I have an instinctive ‘feel’ for the concepts of mass and force, but I have no idea how to measure them and without measuring instructions, no concept in Physics has any practical value (an essential part of most experiments in Physics is the measurement of the values that play a role in the observed process).

How do we measure this inertial mass: m and force: F?

Any time I asked my teachers what mass was, I was told that it is the measure of an object’s inertia (force divided by acceleration). And when I asked what force was, I was again told, predictably, that it was mass times acceleration.

Finally, years later, I read a few books that satisfied my curiosity. One of the best: Richard Feynman’s “Lectures on Physics” recognizes the validity of my question:

“Let us ask, "What is the meaning of the physical laws of Newton, which we write as F = ma? "What is the meaning of force, mass, and acceleration?" Well, we can intuitively sense the meaning of mass, and we can define acceleration if we know the meaning of position and time. We shall not discuss those meanings, but shall concentrate on the new concept of force. The answer is equally simple: "If a body is accelerating, then there is a force on it." That is what Newton's laws say, so the most precise and beautiful definition of force imaginable might simply be to say that force is the mass of an object times the acceleration.”

Then Feynman states the same problem I was struggling with at University:

“If we have discovered a fundamental law, which asserts that the force is equal to the mass times the acceleration, and then define the force to be the mass times acceleration, we have found out nothing.…Now such things certainly cannot be the content of physics, because they are definitions going in a circle….. One might sit in an armchair whole day long and define words at will, but to find out what happens when two balls push against each other or when a weight is hung on a spring, is another matter al¬together, because the way the bodies behave is something completely outside any choice of definitions.”

Finally, Feynman tells us the solution to the dilemma:

“The real content of Newton’s laws is this: that the force is supposed to have some independent properties in addition to the law F=ma; but the specific independent properties that the force has were not completely described by Newton or by anybody else, and therefore the physical law F=ma is an incomplete law.”

As far as “what use is Newton’s Law” is concerned, Feynman states that:

“In order to use Newton’s laws, we have to have some formula for the force; these laws say pay attention to the forces. If an object is accelerating, some agency is at work; find it. Our program for the future of dynamics must be to find the laws for the force. Newton himself went on to give some examples. In the case of gravity he gave a specific formula for the force.”

When I read all this, my philosophical problems were solved: Now I knew what Newton meant and how to use his law to solve practical problems. Once you substitute the formula for the force (be it gravity, tension in springs, etc) then I could easily determine the motion of any object subjected to the force if I knew the mass of the object. "

Zat,

Please be patient with me and correct me where I don't understand.

So...are you saying that in order to understand Force and Mass, then you must also have understood the Force due to Gravity? This actually sounds right to me.

(07-02-2013 08:10 PM)Julius Wrote: So...are you saying that in order to understand Force and Mass, then you must also have understood the Force due to Gravity? This actually sounds right to me.

Julius, gravitational force is one of many forces in nature. It represents the force acting between any two objects that possess mass.

There are many other forces in nature (elastic, electrostatic, magnetostatic, nuclear, etc., etc.)

Newton's second law applies to any force, whatever the source of it is.

What Feynman said was: in order to use the F=m*a equation, we need a formula for the force. He mentioned the gravitational formula of F=G*m1*m2/d^2 -- once you substitute the right side of this equation into F=m*a, then you will have only one unknown left: the mass.

He could have used any other force's formula and have the same result: an equation you can use to measure mass. It does not need to be the gravitational force formula, so you don't need to understand gravitation in order to understand F=m*a.

More from my book on the subject:

"Usually mass is measured by weighing things, but it is not entirely correct conceptually, because weighing an object measures the force of gravitational attraction between the Earth and the body we weigh (gravitational mass). This is conceptually different from the ‘inertial mass’ which is defined as an object’s resistance to acceleration. To make it more clear: just think of how an object’s weight will drastically change (to about one-sixth of its value on Earth) if measured on the Moon, while its inertial mass (resistance to acceleration) will remain the same.

We are not completely defeated though because we have ways to measure inertial mass without weighing the object. For example, by measuring the acceleration of objects under identical forces (e.g. accelerating them by the same spring stretched to the same amount). If the force is identical on two objects (with masses m1 and m2), than F=m1*a1= m2*a2 which means that m1/m2= a2/ a1.

This means that we can measure the ratio of inertial masses between any two objects (by measuring their accelerations under identical forces) and, if we arbitrarily select one particular object as a unit for inertial mass (the prototype of 1kg is a cylinder made of platinum-iridium alloy, kept in a France), then we can measure the inertial mass of any object as described above.

Now that we have a unit of inertial mass (kg) we can define the unit of force as well: 1 unit of force (called a newton) is the force that causes one unit of inertial mass (1 kg) to accelerate at 1 meter per second per second. So now we have a way to measure both inertial masses and forces in their respective units.

Of course, measuring masses by accelerating them with springs is cumbersome and, luckily, we do not have to do that. Instead, as we will soon see, we can take advantage of what seems at first as a lucky coincidence: the inertial and gravitational masses of objects happen to be the same, so we can measure both masses by weighing the objects, after all."

I'm gonna respond to the OP without reading the four pages of responses. You've been warned.

I don't know if you can really 'teach' critical thinking. You can define it, you can have someone read essays and papers such as Charles S. Peirce's Validity of the Laws of Logic (available on 'his' site if you're interested), but some people are just wired differently. In some people the appeal to emotion is stronger than the appeal to logic. As the saying goes, you can lead a horse to water...

Edit: Saw DLJ's post after this one posted and I thought I clear something up for my young American friends. Snooker hall isn't nearly as dirty as it sounds.

Quote: I don't know if you can really 'teach' critical thinking. You can define it, you can have someone read essays and papers ... but some people are
just wired differently. In some people the appeal to emotion is stronger than the appeal to logic. As the saying goes, you can lead a horse to
water...

All skills can be taught and learned - that's the whole point of big-brained animals having a long maturing period. Critical thinking is necessary to all kinds of endeavour. We don't mean formal logic - which is rather tedious and needed only in a narrow speciality. Methods of analysis and investigation can be explained, demonstrated, understood, and practiced, just like painting techniques or using a hacksaw. Not every student will be equally talented or become equally proficient; they can all master the rudiments, just as they can all to read and do sums, though not equally well.

People who are "wired differently" don't function well in the world and need more help. That just means we have to find ways of reaching them, figure out what they are capable of; how they communicate and learn. I think, though, we were talking about regular students, IQ 100-130.

As for various proclivities and talents, whether somebody is more visual than verbal, more empathic than calculating, more intuitive than deductive, it doesn't matter. They all have to grow up, fit into a society, take on the responsibilities of citizenship. One baby likes peaches, the other likes pears, both can learn to use a spoon.

(07-02-2013 11:44 PM)Peterkin Wrote: All skills can be taught and learned - that's the whole point of big-brained animals having a long maturing period. Critical thinking is necessary to all kinds of endeavour. We don't mean formal logic - which is rather tedious and needed only in a narrow speciality. Methods of analysis and investigation can be explained, demonstrated, understood, and practiced, just like painting techniques or using a hacksaw. Not every student will be equally talented or become equally proficient; they can all master the rudiments, just as they can all to read and do sums, though not equally well.

People who are "wired differently" don't function well in the world and need more help. That just means we have to find ways of reaching them, figure out what they are capable of; how they communicate and learn. I think, though, we were talking about regular students, IQ 100-130.

As for various proclivities and talents, whether somebody is more visual than verbal, more empathic than calculating, more intuitive than deductive, it doesn't matter. They all have to grow up, fit into a society, take on the responsibilities of citizenship. One baby likes peaches, the other likes pears, both can learn to use a spoon.

Power rules, usually this means majority rules. Clearly you can thrive without this whether it is a beneficial trait or not. Just sayin'...

(07-02-2013 08:10 PM)Julius Wrote: So...are you saying that in order to understand Force and Mass, then you must also have understood the Force due to Gravity? This actually sounds right to me.

Julius, gravitational force is one of many forces in nature. It represents the force acting between any two objects that possess mass.

There are many other forces in nature (elastic, electrostatic, magnetostatic, nuclear, etc., etc.)

Newton's second law applies to any force, whatever the source of it is.

What Feynman said was: in order to use the F=m*a equation, we need a formula for the force. He mentioned the gravitational formula of F=G*m1*m2/d^2 -- once you substitute the right side of this equation into F=m*a, the you will have only one unknown left: the mass.

He could have used any other force's formula and have the same result: an equation you can use to measure mass. It does not need to be the gravitational force formula, so you don't need to understand gravitation in order to understand F=m*a.

More from my book on the subject:

"Usually mass is measured by weighing things, but it is not entirely correct conceptually, because weighing an object measures the force of gravitational attraction between the Earth and the body we weigh (gravitational mass). This is conceptually different from the ‘inertial mass’ which is defined as an object’s resistance to acceleration. To make it more clear: just think of how an object’s weight will drastically change (to about one-sixth of its value on Earth) if measured on the Moon, while its inertial mass (resistance to acceleration) will remain the same.

We are not completely defeated though because we have ways to measure inertial mass without weighing the object. For example, by measuring the acceleration of objects under identical forces (e.g. accelerating them by the same spring stretched to the same amount). If the force is identical on two objects (with masses m1 and m2), than F=m1*a1= m2*a2 which means that m1/m2= a2/ a1.

This means that we can measure the ratio of inertial masses between any two objects (by measuring their accelerations under identical forces) and, if we arbitrarily select one particular object as a unit for inertial mass (the prototype of 1kg is a cylinder made of platinum-iridium alloy, kept in a France), then we can measure the inertial mass of any object as described above.

Now that we have a unit of inertial mass (kg) we can define the unit of force as well: 1 unit of force (called a newton) is the force that causes one unit of inertial mass (1 kg) to accelerate at 1 meter per second per second. So now we have a way to measure both inertial masses and forces in their respective units.

Of course, measuring masses by accelerating them with springs is cumbersome and, luckily, we do not have to do that. Instead, as we will soon see, we can take advantage of what seems at first as a lucky coincidence: the inertial and gravitational masses of objects happen to be the same, so we can measure both masses by weighing the objects, after all."

Zat,

I think I am starting to see where you are coming from. You surely make more sense than me. Please allow me some time to think.